STOCHASTIC

36 LELAND
An alternative approach to proving the existence of optimal policies
is to assign further properties to the preference ranking over actions, as
reflected by the objective function. If these properties alone ensure
existence, there is no need to incur the costs associated with bounding
the action set.
This paper considers a class of decision problems encompassing several
of relevance to economists, including the choice of optimal portfolios
and the choice of optimal inputs and outputs by a perfectly competitive
firm (with constant or decreasing returns to scale) facing random prices.
Weak, and indeed commonly assumed, properties of the utility function
are shown to ensure the existence of optimal policies. Perhaps the most
interesting implication of the results is that the perfectly competitive firm
with constant returns to scale will have a determinate output when there
is uncertainty regarding prices, and it has a risk-averse utility function
over profits which is bounded above.
II. A CLASS OF DECISION PROBLEMS
The structure of a typical decision problem includes a set S of states
of nature, a o--algebra Sf of events, a set C of outcomes (hereafter assumed
to be in R 1 ), and a set D of decisions, each element of which.is characterized
by a measurable function mapping 5 into C. In all decision
problems, it is assumed that the decision-maker possesses a preference
ranking over the elements of D. An optimal policy or action is defined
as d* e D such that d* ~>d for all de D. Of course, such an action may
not exist.
The problems examined in subsequent sections are a special class of
the general decision problem. As above, elements include a set of states
of nature and associated u-algebra of events, and a set CC R 1 of outcomes.
But the actions belong to a set X of feasible gambling positions in a set G
of gambling opportunities.
Each element g of the gambling opportunity set G is characterized by
a measurable net return function mapping S into a set of monetary
returns C" C R 1 . g(s), therefore, is the monetary return per unit of the
gamble g, depending on the state of nature. If g(s) = k for all seS, the
"gamble" provides a sure return k. We assume the following:
CI G has a finite number of elements, g l ,...,g".
G.2 G is independent of the actions of the decision-maker.
G.l seems warranted, at least in the static case, by the finite number of
stocks, bonds, products, etc., in which one can invest. G.l is a straight-
PART III. STATIC PORTFOLIO SELECTION MODELS